Nephroid
Rolling curves (roulettes)The two-cusped epicycloid — the bright caustic you see when sunlight reflects inside a cup.
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Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Nephroid?
The nephroid — “kidney-shaped” — is the epicycloid with two cusps, traced by a circle rolling around a fixed circle of twice its radius. Proctor named it in 1878, though Huygens had already connected it to optics in 1678.
It is the everyday caustic: when parallel rays (sunlight) reflect off the inside of a circular cup, the reflected rays envelope one half of a nephroid — the bright cusped line on the surface of your coffee. A point source on the rim gives a cardioid instead; the two curves are cousins that answer slightly different lighting setups.
Key properties
- Arc length 24r and area 12πr².
- It is the epicycloid with k = 2 (R = 2r).
- Caustic of a circle under parallel light — the coffee-cup curve.
- Its evolute is another nephroid, half the size and rotated 90°.
- The name comes from the Greek nephros, “kidney”.
Where you'll see it
Coffee-cup light caustics, reflector design, and epicycloid families in gear geometry.